
Similar to the treatment of term \( Q \)  for the Navier-Stokes equations (cf. [1]}), we can get
\(\left| \int _{\mathbb {R}} Q \,\mathrm {d}x \right| \le C \delta (1+t)^{-\frac{5}{4}} + C (\eta + \varepsilon _0) \Vert  (\psi _x,\zeta _x) \Vert ^2 + C \delta \int _{\mathbb {R}} (\phi ^2 + \zeta ^2) \omega ^2 \,\mathrm {d}x.\) 
